- Screening
- Pros: early detection
4/5/2021
https://howardwolinsky.substack.com/p/the-big-lie-and-low-risk-prostate?s=r
Gleason score
3 + 3 (6): low risk
≥ 3 + 4/4 + 3 (≥7): intermediate risk
Drawback
Painful
Infections, sepsis
How can we make biopsy schedules in AS on an individual level?
How can our personalized schedules relieve the burden of biopsies in AS at acceptable cost?
Tomer, A., Nieboer, D., Roobol, M. J., Steyerberg, E. W., & Rizopoulos, D. (2022). Shared decision making of burdensome surveillance tests using personalized schedules and their burden and benefit. Statistics in medicine, 10.1002/sim.9347.
Target population: Patients diagnosed as Prostate cancer with Gleason score of 3+3
Sample size: 850 subjects recorded in the dataset, among which 833 participants had adequate PSA measurements for analysis.
Data types:
Repeated measurements: PSA values over time
Time-to-event data: time until progression
baseline variables: age at diagnosis and PSA density
Competing risk: treatment before progression (87/833)
Interval censoring: periodical examinations
\[ \begin{align} \log_2(\text{PSA}_i + 1)(t) &= \color{blue}{m_i(t)} + \epsilon_i(t)\\ \color{blue}{m_i(t)} &= \beta_0 + b_0 + \underbrace{(\beta_1 + b_1)\mathcal{C}^{(1)}_i(t) + (\beta_2 + b_2)\mathcal{C}^{(2)}_i(t) + (\beta_3 + b_3)\mathcal{C}^{(3)}_i(t)}_{\text{natural cubic splines for time variable}} + \underbrace{\beta_4(\text{Age}_i - 62)}_{\text{baseline covariate}} \end{align} \] where \(\mathcal{C}(t)\)is the design matrix for the natural cubic splines for time \(t\); the error term \(\epsilon_i(t) \sim t_3\)
The hazard, \(h_i^{(k)}(t)\), of experiencing event \(k\) (\(k = 1\) indicates progression; \(k = 2\) indicates treatment) for subject \(i\) at time \(t\) is: \[ \begin{align*} h_i^{(k)}(t \mid \boldsymbol{\mathcal{M}}_i(t)) = h_0^{(k)}(t)\exp\Big\{\gamma_k\text{density}_i + \alpha_{1k} \color{blue}{m_i(t)} + \alpha_{2k}\big[\color{blue}{m_i(t)} - \color{blue}{m_i(t-1)}\big]\Big\} \end{align*} \] where:
\(\boldsymbol{\mathcal{M}}_i(t)\) is the estimated true history of PSA levels for subject \(i\)
\(\color{blue}{m_i(t)}\) is the estimate of the true PSA level for subject \(i\) at \(t\)
\(\color{blue}{m_i(t)} - \color{blue}{m_i(t-1)}\) is the estimate of the PSA level change in the previous year from \(t\) for subject \(i\)
More interest goes to the probability of getting progression, which can be achieved by cause-specific cumulative incidence function
\[\text{CIF}_j^{k=1}(t) = \int^t_0 \underbrace{h_j^{(1)}(s)}_{\text{instantaneous risk}\\ \text{of progression}}\underbrace{\exp\left\{-\sum^K\int^s_0h_j^{(k)}(\nu)d\nu\right\}}_{\text{overall survival function}}\]
Two indicators
Number of biopsies to be conducted
Detection delay
Impact of detection delay on the patients?
Progression-specific risk
Assume precision of biopsies
\[ \begin{align*} \mathit{Nb}_i(S_i^{\kappa}) = \begin{cases} 1, &t < T^*_i \leq s_1, k=1 \\ 2, &s_1 < T^*_j \leq s_2, k =1 \\ ... \\ N_i, &s_{N_i - 1} < T^*_i \leq s_{N_i}, k=1 \end{cases} \end{align*} \]
Generally, the expected number of biopsies:
\[ \begin{align*} E\{\mathit{Nb}_i(S_i^\kappa)\} = \sum_{n=1}^{N_i} n \times \underbrace{\Pr\{s_{n-1}<T^*_i \leq s_n | T^*_i \leq s_{N_i}, k =1 \}}_{\text{probability of progressing between two biopsies}}, \ \ s_0 = t \end{align*} \]
List all the possibilities of occurrence of progression:
\[ \begin{align*} \mathit{Dd}_j(S_j^{\kappa}) = \begin{cases} s_1 - T^*_i, &t < T^*_i \leq s_1, k=1 \\ s_2 - T^*_i, &s_1 < T^*_i \leq s_2, k =1 \\ ... \\ s_{N_i} - T^*_i, &s_{N_i - 1} < T^*_i \leq s_{N_i}, k =1 \end{cases} \end{align*} \]
Generally, the expected detection delay is:
\[ \begin{align*} E\{\mathit{Dd}_j(S_i^\kappa)\} = \sum_{n=1}^{N_i} \big[s_n - \underbrace{E(T^*_i | s_{n-1}, s_{n}, v, k=1)}_{\text{expected progression time}}\big] \times \underbrace{\Pr\{s_{n-1}<T^*_i \leq s_n | T^*_i \leq s_{N_i}, k =1 \}}_{\text{probability of progressing between two biopsies}} \end{align*} \]